Ok, have investigated a bit more.
Should've/could've mentioned it earlier, no clue why I didn't, but though technically possible, using the real name (not "username") is a no-no, of course, because it's not unique: two different users can have the same name, at least in the real world and, unless you made very significant changes (don't!), also in Joomla and CB.
Once we talk names, username is unique in the system, so that'll remain your best bet. And it's already the default for URLs, so no changes needed.
Anyhow, as indicated in my post above, I'm generally set for "emails as usernames" for users not needing to remember usernames and to log in with their email, promptly resulting in "ugly" URLs.
Except, I made since one small code change, and now my profile URLs are containing the user ID instead. Both client and me are happy with that.
(In case anyone wonders: the change was small but in a core file, so no posting here in respect for JP's policies.)